A note on the optimal boundary regularity for the planar generalized $p$-Poisson equation

TL;DR

This paper proves sharp boundary regularity results for solutions to a generalized p-Poisson equation in the plane, extending interior regularity findings to boundary cases with optimal assumptions on data and boundary conditions.

Contribution

It establishes boundary regularity for the generalized p-Poisson equation with sharp conditions, matching the linear case, and extends interior regularity results to boundary scenarios.

Findings

Sharp boundary regularity results for solutions in the plane.

Regularity assumptions are optimal and match linear case.

Extension of interior regularity to boundary conditions.

Abstract

In this note, we establish sharp regularity for solutions to the following generalized $p$- Poisson equation $$-\ div\ \big(\langle A\nabla u,\nabla u\rangle^{\frac{p-2}{2}}A\nabla u\big)=-\ div\ \mathbf{h}+f$$ in the plane (i.e. in $\mathbb{R}^n=\mathbb{R}^2$) for $p>2$ in the presence of Dirichlet as well as Neumann boundary conditions and with $\mathbf{h}\in C^{1-n/q}$, $f\in L^q$, $2=n<q\leq\infty$. The regularity assumptions on the principal part $A$ as well as that on the Dirichlet/Neumann conditions are exactly the same as in the linear case and therefore sharp (see Remark 2.5 below). Our main results Theorem 2.3 and Theorem 2.4 should be thought of as the boundary analogues of the sharp interior regularity result established in the recent interesting paper [1] in the case of \begin{equation}\label{e0} -\ div\ (|\nabla u|^{p-2} \nabla u) =f \end{equation} for more general variable coefficient operators and with an additional divergence term.